The k p Method: Electronic Properties of Semiconductors

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A number of new and interesting results have been obtained.


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The computational implementation using the finite difference method of the Burt-Foreman theory for two dimensional nanostructures has confirmed that a non-uniform grid is much more efficient, as had been obtained by others in one dimensional nanostructures. In addition we have demonstrated that the multiband problem can be very effectively and efficiently solved with commercial software FEMLAB. Two of the most important physical results obtained and discussed in the dissertation are the following.

One is the first ab initio demonstration of possible electron localization in a nanowire superlattice in a barrier material, using a full numerical solution to the one band kp equation. The second is the demonstration of the exactness of the Sercel-Vahala transformation for cylindrical wurtzite nanostructures. Comparison of the subsequent calculations to experimental data on CdSe nanorods revealed the important role of the linear spin splitting term in the wurtzite valence band.

All authors have granted to WPI a nonexclusive royalty-free license to distribute copies of the work. Copyright is held by the author or authors, with all rights reserved, unless otherwise noted. Three different forms are presented, namely the E k digram, the E k diagram combined with the reduced-zone diagram as well as the reduced-zone diagram only. From Figure 2. The range of energies for which there is no solution is referred to as an energy band gap. These correspond to local minima and maxima of the E k relation. The reduced-zone diagram shown in Figure 2.

For instance the second energy bandgap occurs between 1. Complete energy band diagrams of semiconductors are very complex. However, most have features similar to that of the diamond crystal discussed in section 2. In this section, we first take a closer look at the energy band diagrams of common semiconductors. We then present a simple diagram containing only the most important features and discuss the temperature and doping dependence of the energy bandgap.

The energy band diagrams of semiconductors are rather complex. The detailed energy band diagrams of germanium, silicon and gallium arsenide are shown in Figure 2. The energy is plotted as a function of the wavenumber, k , along the main crystallographic directions in the crystal, since the band diagram depends on the direction in the crystal. The energy band diagrams contain multiple completely-filled and completely-empty bands. In addition, there are multiple partially-filled band. Fortunately, we can simplify the energy band diagram since only the electrons in the highest almost-filled band and the lowest almost-empty band dominate the behavior of the semiconductor.

The energy band diagrams shown in the previous section are frequently simplified when analyzing semiconductor devices. Since the electronic properties of a semiconductor are dominated by the highest partially empty band and the lowest partially filled band, it is often sufficient to only consider those bands. This leads to a simplified energy band diagram for semiconductors as shown in Figure 2. The diagram identifies the almost-empty conduction band by a set of horizontal lines. The bottom line indicates the bottom edge of the conduction band and is labeled E c. Similarly, the top of the valence band is indicated by a horizontal line labeled E v.

The energy band gap, E g , is located between the two bands. The distance between the conduction band edge, E c , and the energy of a free electron outside the crystal called the vacuum level labeled E vacuum is quantified by the electron affinity, c multiplied with the electronic charge q.

Chapter 2: Semiconductor Fundamentals

An important feature of an energy band diagram, which is not included on the simplified diagram, is whether the conduction band minimum and the valence band maximum occur at the same value for the wavenumber. If so, the energy bandgap is called direct. If not, the energy bandgap is called indirect. This distinction is of interest for optoelectronic devices since direct bandgap materials provide more efficient absorption and emission of light.

For instance, the smallest bandgap of germanium and silicon is indirect, while gallium arsenide has a direct bandgap as can be seen on Figure 2. The energy bandgap of semiconductors tends to decrease as the temperature is increased. This behavior can be understood if one considers that the interatomic spacing increases when the amplitude of the atomic vibrations increases due to the increased thermal energy.

k p Method - E-bok - Lok C Lew Yan Voon, Morten Willatzen () | Bokus

This effect is quantified by the linear expansion coefficient of a material. An increased interatomic spacing decreases the average potential seen by the electrons in the material, which in turn reduces the size of the energy bandgap. A direct modulation of the interatomic distance - such as by applying compressive tensile stress - also causes an increase decrease of the bandgap. The temperature dependence of the energy bandgap, E g , has been experimentally determined yielding the following expression for E g as a function of the temperature, T :.

These fitting parameters are listed for germanium, silicon and gallium arsenide in Table 2. A plot of the resulting bandgap versus temperature is shown in Figure 2. The bandgap of silicon at K equals:.


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Similarly one finds the energy bandgap for germanium and gallium arsenide, as well as at different temperatures, yielding:. High doping densities cause the bandgap to shrink. This effect is explained by the fact that the wavefunctions of the electrons bound to the impurity atoms start to overlap as the density of the impurities increase.

For instance, at a doping density of 10 18 cm -3 , the average distance between two impurities is only 10 nm. This overlap forces the energies to form an energy band rather than a discreet level. If the impurity level is shallow see section 2. From this expression we find that the bandgap shrinkage can typically be ignored for doping densities less than 10 18 cm A plot of the change in bandgap energy with doping density is shown in Figure 2.

Once we know the bandstructure of a given material we still need to find out which energy levels are occupied and whether specific bands are empty, partially filled or completely filled.

Introduction to Semiconductors

Empty bands do not contain electrons. Therefore, they are not expected to contribute to the electrical conductivity of the material. Partially filled bands do contain electrons as well as available energy levels at slightly higher energies. These unoccupied energy levels enable carriers to gain energy when moving in an applied electric field. Electrons in a partially filled band therefore do contribute to the electrical conductivity of the material. Completely filled bands do contain plenty of electrons but do not contribute to the conductivity of the material.

This is because the electrons cannot gain energy since all energy levels are already filled. In order to find the filled and empty bands we must find out how many electrons can be placed in each band and how many electrons are available. Each band is formed due to the splitting of one or more atomic energy levels. Therefore, the minimum number of states in a band equals twice the number of atoms in the material. The reason for the factor of two is that every energy level can contain two electrons with opposite spin.

To further simplify the analysis, we assume that only the valence electrons the electrons in the outer shell are of interest. The core electrons are tightly bound to the atom and are not allowed to freely move in the material.

The k p Method: Electronic Properties of Semiconductors

A half-filled band is shown in Figure 2. This situation occurs in materials consisting of atoms, which contain only one valence electron per atom. Most highly conducting metals including copper, gold and silver satisfy this condition. Materials consisting of atoms that contain two valence electrons can still be highly conducting if the resulting filled band overlaps with an empty band. This scenario is shown in b.

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No conduction is expected for scenario d where a completely filled band is separated from the next higher empty band by a larger energy gap. Such materials behave as insulators. Finally, scenario c depicts the situation in a semiconductor. The completely filled band is now close enough to the next higher empty band that electrons can make it into the next higher band.

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This yields an almost full band below an almost empty band. Lyssna fritt i 30 dagar! Ange kod: play Du kanske gillar. Ladda ned.